On the boundedness and Schatten class property of noncommutative martingale paraproducts and operator-valued commutators

TL;DR

This paper investigates the Schatten class membership and boundedness of noncommutative martingale paraproducts and operator-valued commutators, introducing new techniques that extend previous results to more general operators and complex kernels.

Contribution

It develops a new approach using Hytönen's dyadic martingale technique and the complex median method, enabling analysis of more general singular integral operators and complex-valued kernels in noncommutative settings.

Findings

Characterizes Schatten class membership for noncommutative martingale paraproducts.

Provides sufficient conditions for boundedness of operator-valued commutators.

Introduces a new approach applicable to complex-valued kernels and semicommutative settings.

Abstract

We study the Schatten class membership of semicommutative martingale paraproducts and use the transference method to describe Schatten class membership of purely noncommutative martingale paraproducts, especially for CAR algebras and $\mathop{\otimes}\limits_{k=1}^{\infty}\mathbb{M}_{d}$ in terms of martingale Besov spaces. Using Hyt\"{o}nen's dyadic martingale technique, we also obtain sufficient conditions on the Schatten class membership and the boundedness of operator-valued commutators involving general singular integral operators. We establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the Schatten class membership of operator-valued commutators associated with non-degenerate kernels in Hyt\"{o}nen's sense. This resolves the problem on the characterization of Schatten class membership of operator-valued commutators. Our results are new even in the scalar case. Our new approach is built on Hyt\"{o}nen's dyadic martingale technique and the complex median method. Compared with all the previous ones, this new one is more powerful in several aspects: $(a)$ it permits us to deal with more general singular integral operators with little smoothness; $(b)$ it allows us to deal with commutators with complex-valued kernels; $(c)$ it goes much further beyond the scalar case and can be applied to the semicommutative setting. By a weak-factorization type decomposition, we get some necessary but not optimal conditions on the boundedness of operator-valued commutators. In addition, we give a new proof of the boundedness of commutators still involving general singular integral operators concerning $BMO$ spaces in the commutative setting.